Problem: Simplify and expand the following expression: $ \dfrac{2q - 6}{2q + 1}+\dfrac{q + 6}{4q - 10} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2q + 1)(4q - 10)$ Multiply the first term by $\dfrac{4q - 10}{4q - 10}$ $ \begin{align*} \dfrac{2q - 6}{2q + 1} \times \dfrac{4q - 10}{4q - 10} & = \dfrac{(2q - 6)(4q - 10)}{(2q + 1)(4q - 10)} \\ & = \dfrac{8q^2 - 44q + 60}{(2q + 1)(4q - 10)}\end{align*} $ Multiply the second term by $\dfrac{2q + 1}{2q + 1}$ $ \begin{align*} \dfrac{q + 6}{4q - 10} \times \dfrac{2q + 1}{2q + 1} & = \dfrac{(q + 6)(2q + 1)}{(4q - 10)(2q + 1)} \\ & = \dfrac{2q^2 + 13q + 6}{(4q - 10)(2q + 1)}\end{align*} $ Now we have: $ = \dfrac{8q^2 - 44q + 60}{(2q + 1)(4q - 10)} + \dfrac{2q^2 + 13q + 6}{(4q - 10)(2q + 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{8q^2 - 44q + 60 + 2q^2 + 13q + 6}{(2q + 1)(4q - 10)} $ $ = \dfrac{10q^2 - 31q + 66}{(2q + 1)(4q - 10)}$ Expand the denominator: $ = \dfrac{10q^2 - 31q + 66}{8q^2 - 16q - 10}$